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A process conception of function has become widely recognized as crucial for understanding topics in collegiate mathematics, and several researchers have sought to characterize ways of thinking about functions that support a strong process conception (e.g., Carlson, 1998; Carlson et al., 2002; Oehrtman, 2008; Thompson, 1994). One way of thinking about func- tions that has been shown to be favorable for students developing an image of function as a dynamic process is imagining the covariation (simultaneous change) of two quantities.

The notion of covariation first appeared in mathematics education literature by Con- frey and Smith (1991, 1994, 1995). They described covariation by stating, “as one quantity changes in a predictable or recognizable pattern, the other also changes, typically in a dif- fering pattern” (Confrey & Smith, 1991, p. 57). In a later paper, they described covariation as “the juxtaposition of two sequences, each of which is generated independently through a pattern of data values” (Confrey & Smith, 1995, p. 67). In addition to being able to describe the pattern of each sequence individually, students with a strong understanding of covariation can coordinate changes in values of one sequence with changes in values of the other. Covariation is a more general description of the function concept and can be used to describe one-to-many correspondences, which are typically rejected in a traditional correspondence approach to functions. For this reason, Confrey and Smith advocate a more balanced approach to the study of functions that incorporates both covariation and the traditional correspondence approach.

A natural representation of covariation is a table that enables an individual to coordinate changes in values from one column with changes in values from the other. However, such a representation is of discrete covariation. Saldanha and Thompson (1998) hypothesized that students, through graphing, can construct an image of continuous covariation. The authors described continuous covariation as follows:

In this regard, our notion of covariation is of someone holding in mind a sustained image of two quantities (magnitudes) simultaneously. It entails coupling the two quantities, so that, in one’s understanding, a multiplicative object is formed of the two. As a multiplicative object, one tracks either quantity’s value with the immediate, explicit, and persistent realization that, at every moment, the other quantity also has a value. (Saldanha & Thompson, 1998, p. 298)

Saldanha and Thompson (1998) documented the problem-solving of an 8th-grade stu- dent, Shawn, during a teaching experiment designed to support thinking about continuous covariation. The tasks in the teaching experiment were graphing activities that usedGeome- ter’s Sketchpad and involved tracking and describing the behavior of the distances between a car and two cities as it moves along a road. A sliding point C, representing the car, was placed on a segment, which represented the road, and two fixed pointsAandB, representing

the two cities, were placed off of the segment. Then the varying distances of C from A and B, respectively, were tracked on the vertical and horizontal axes. The tasks of this teaching experiment were administered in the three following phases:

• Engagement—which allowed Shawn to explore each distance individually, each rep- resented by line segments—one on a vertical axis and one on a horizontal axis—of varying length, and then both distances simultaneously, which were represented by a moving pointP and its locus in the plane. Shawn was said to have internalized these actions when he couldimagine moving the car and describe the anticipated effects on the point P.

• Move to representation—which supported Shawn’s internalization of covariation. He was presented with various road-city configurations and was asked to imagine moving the car and describe the effect it would have on the point P.

• Move to reflection—which was intended to have Shawn come to imagine completed covariation. He was presented with various graphs of P’s locus and was asked to predict the locations of the two cities relative to the road.

The results of the study by Saldanha and Thompson (1998) indicate that it is non- trivial for students to understand graphs as representing a continuum of states of covarying quantities.

Based on the notions of covariation of Confrey and Smith (1994, 1995), Saldanha and Thompson (1998), and others, Carlson et al. (2002) proposed a framework for investigating covariational reasoning employed by students when describing dynamic events. The authors defined covariational reasoning to be “the cognitive activities involved in coordinating two varying quantities while attending to the ways in which they change in relation to each other” (Carlson et al., 2002, p. 354). Their framework consists of five mental actions (MA) and associated behaviors, which are described in Table 2.1. The mental actions are not hierarchical—it is possible, for example, for a student to exhibit MA5 reasoning without exhibiting each of MA1–MA4 reasoning. However, the covariation framework is hierarchical

Table 2.1 Mental actions of the covariation framework (Carlson et al., 2002). Mental action Description of mental action Behaviors

Mental Action 1 (MA1)

Coordinating the value of one vari- able with changes in the other.

Labeling the axes with verbal indica- tions of coordinating the two variables (e.g., y changes with changes in x). Mental Action 2

(MA2)

Coordinating the direction of change of one variable with changes in the other variable.

Constructing an increasing straight line; Verbalizing an awareness of the direc- tion of change of the output while con- sidering changes in the input.

Mental Action 3 (MA3)

Coordinating the amount of change of one variable with changes in the other variable.

Plotting point/constructing secant lines; Verbalizing an awareness of the amount of change of the output while considering changes in the input. Mental Action 4

(MA4)

Coordinating the average rate-of- change of the function with uni- form increments of change in the input variable.

Constructing contiguous secant lines for the domain; Verbalizing an awareness of the rate of change of the output (with respect to the input) while considering uniform increments of the input. Mental Action 5

(MA5)

Coordinating the instantaneous rate of change of the function with continuous changes in the indepen- dent variable for the entire domain of the function.

Constructing a smooth curve with clear indications of concavity changes; Ver- balizing an awareness of the instanta- neous changes in the rate of change for the entire domain of the function (direc- tion of concavities and inflection points are correct).

by associating a level of reasoning with a student who is able to perform a certain mental action and the mental actions with a lower number. Most students3 in the study by Carlson et al. (2002) exhibited all of the mental actions MA1–MA3, while students who demonstrated MA5 reasoning were unable to “unpack” the mental action in terms of MA1–MA4.

Carlson et al. (2002) concluded their study by recommending that students have the opportunity to think about covariation in real-world problems. They further suggested that, although their study focused on graphing tasks and visual imagery, covariational reason- ing should be extended to working with other representations of functions. Additionally,

they suggested that future research consider how students use the implicit time variable in covariational reasoning.

Oehrtman et al. (2008) asserted that a process conception of function (in the sense of APOS) is foundational for covariational reasoning. According to them, a student with a strong process conception can view a formula as a way of transforming an input value into an output value, and, by transiting between representations, the student can begin to explore how increases in the input value effect the output value. As stated by Carlson et al. (2002, p. 376), “[. . .] the covariation framework may be used to infer information not just about the developmental level of student images of covariation but also about the internal structure of these images.” In other words, investigating students’ covariational reasoning abilities can provide a more refined description of their process conception of function.

As previously stated, Saldanha and Thompson (1998) described continuous covariation as the mentally pairing of two covarying quantities to create a multiplicative object. Then “one tracks either quantity’s value with the immediate, explicit, and persistent realization that, at every moment, the other quantity also has a value” (Saldanha & Thompson, 1998, p. 298). In a much later paper, Thompson (2011) reformulated this description mathemat- ically to capture the dynamic aspects of covariation. He accomplished this by utilizing the notion of covering and by making timet an explicit variable. First, the set of all values of t is covered by intervals of size: [t, t+). Then, astis varied, the interval coverings represent variation in chunks, and suggests that these chunks are infinitesimally small. Therefore, if t = [t, t+), then (x(t), y(t)) represents two quantities x and y continuously covarying as a single entity (x, y) as time passes. This is comparable to Thompson’s (1994) earlier distinction between the notions of ratio and rate. He defined a ratio to be “the result of comparing two quantities multiplicatively” (p. 190). Thompson further explained that when a ratio is reconceived as applying “outside of the phenomenal bounds in which it was origi- nally conceived, then one has generalized that ratio to a rate” (1994, p. 192). In other words, two quantities are paired by some quantitative operation, and then the relationship between the quantities is perceived as invariant as the they simultaneously change with respect to

time.

Castillo-Garsow (2010, 2011) distinguished between two types of reasoning about con- tinuous variation—chunky reasoning and smooth reasoning. Chunky reasoning is thinking about change in completed chunks of time, whereas smooth reasoning is thinking about change in progress. As asserted by Castillo-Garsow (2011), chunky reasoning is inherently discrete and, therefore, counter to continuous variation, since it inhibits the ability to imag- ine change in real time. In order to employ chunky reasoning to consider variation within a chunk of time, the chunk must be re-conceptualized into smaller chunks. Smooth reasoning, on the other hand, is inherently continuous. It entails imagining passing through every mo- ment within a larger unit of time. Castillo-Garsow (2010, p. 230) interpreted Confrey and Smith’s (1994, 1995) perspective of covariation as chunky reasoning and Thompson’s (2011) perspective as a mixture of chunky and smooth reasoning.

The results of Castillo-Garsow’s dissertation (2010) showed that one student, Derek, demonstrated proficiency in both chunky and smooth reasoning when thinking about situ- ations involving exponential relationships, and, when appropriate, Derek adopted one way of reasoning over the other. Another student, Tiffany, was limited mostly to chunky rea- soning, and it put her at a disadvantage when attempting to describe the overall behavior of functions. For this reason, Castillo-Garsow claimed that chunky reasoning can suffice for numerical calculations, but students whose thinking about covariation is limited to chunky reasoning might not be able to accomplish other tasks. Therefore, it is necessary for students to develop both chunky and smooth reasoning abilities.

Johnson (2013) distinguished between two ways of reasoning about quantities varying together—change-dependent reasoning and simultaneous-independent reasoning. Change- dependent reasoning is characterized by treating one quantity as the independent variable and the other as the dependent variable, so that a change in the second quantity is in response to a change in the first quantity, whereas simultaneous-independent reasoning is characterized by viewing each quantity as changing independently of the other in relation to time. In one report, Johnson (2012b) documented her observations of a student—Hannah—

using change-dependent reasoning, while in another report, Johnson (2012a) documented her observations of three students—Austin, Mason, and Jacob—using simultaneous-independent reasoning. Hannah was able to use change-dependent reasoning to describe variation in the rate-of-change of two quantities, while Austin, Mason, and Jacob’s simultaneous-independent reasoning only enabled them to compare amounts of change in two quantities. She concluded, “such comparisons merely related quantities through division rather than representing a newly constructed quantity” (Johnson, 2012a, p. 52).